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Wind wave
In fluid dynamics, wind waves, or windgenerated waves, are surface waves that occur on the free surface of oceans, seas, lakes, rivers, and canals or even on small puddles and ponds. They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves range in size from small ripples, to waves over 100 ft (30 m) high.^{[1]}
When directly generated and affected by local winds, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells. More generally, a swell consists of windgenerated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago.^{[2]} Wind waves in the ocean are called ocean surface waves.
Wind waves have a certain amount of randomness: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a stochastic process, in combination with the physics governing their generation, growth, propagation and decay—as well as governing the interdependence between flow quantities such as: the water surface movements, flow velocities and water pressure. The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models.
Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have winddriven waves.^{[3]}
Contents
Wave formation
The great majority of large breakers seen on a beach result from distant winds. Five factors influence the formation of the flow structures in wind waves:^{[4]}
 Wind speed or strength relative to wave speed the wind must be moving faster than the wave crest for energy transfer
 The uninterrupted distance of open water over which the wind blows without significant change in direction (called the fetch)
 Width of area affected by fetch
 Wind duration  the time over which the wind has blown over a given area
 Water depth
All of these factors work together to determine the size of wind waves and the structures of the flows within:
 Wave height (from high trough to crest)
 Wave length (from crest to crest)
 Wave period (time interval between arrival of consecutive crests at a stationary point)
 Wave propagation direction
• Sources of wind wave generation: Sea water wave is generated by many kinds of disturbances such as Seismic events, gravity, and crossing wind. The generation of wind wave is initiated by the disturbances of cross wind field on the surface of the sea water. Two major Mechanisms of surface wave formation by winds (a.k.a.‘The MilesPhillips Mechanism’) and other sources (ex. earthquakes) of wave formation can explain the generation of wind waves.
However, if one set a flat water surface (Beaufort Point,0) and abrupt cross wind flows on the surface of the water,then the generation of surface wind waves can be explained by following two mechanisms which initiated by normal pressure fluctuations of turbulent winds and parallel wind shear flows.
• The mechanism of the surface wave generation by winds
1) Starts from "Fluctuations of wind" (O.M.Phillips) : the wind wave formation on water surface by wind is started by a random distribution of normal pressure acting on the water from the wind. By the mechanism developed by O.M. Phillips (in 1957), the water surface is initially at rest and wave generation is started by adding turbulent wind flows and then, by the fluctuations of the wind, normal pressure acting on the water surface. From this pressure fluctuation arise normal and tangential stresses to the surface water, which generates wave behavior on the water surface. It is assumed that: ^{[6]}
 The water originally at rest.
 The water is not viscid.
 The water is irrotational.
 There are random distribution of normal pressure to the water surface from the turbulent wind.
 Correlations between air and water motions are neglected.
2) starts from "wind shear forces" on the water surface (J.W.Miles, applied to mainly 2D deep water gravity waves) ; John W. Miles suggested a surface wave generation mechanism which is initiated by turbulent wind shear flows Ua(y), based on the inviscid OrrSommerfeld equation in 1957. He found the energy transfer from wind to water surface as a wave speed, c is proportional to the curvature of the velocity profile of wind Ua’’(y) at point where the mean wind speed is equal to the wave speed (Ua=c, where, Ua is the Mean turbulent wind speed). Since the wind profile Ua(y) is logarithmic to the water surface, the curvature Ua’’(y) have negative sign at the point of Ua=c. This relations show the wind flow transferring its kinetic energy to the water surface at their interface, and arises wave speed, c.
the growthrate can be determined by the curvature of the winds ((d^2 Ua)/(dz^2 )) at the steering height (Ua (z=z_h)=c) for a given wind speed Ua {Assumptions; 1. 2D parallel shear flow, Ua(y) 2. incompressible, inviscid water / wind 3. irrotational water 4. slope of the displacement of surface is small}^{[7]}
Generally these wave formation mechanisms occur together on the ocean surface and arise wind waves and grows up to the fully developed waves.
For example,^{[8]}
If we suppose a very flat sea surface (Beaufort number, 0), and sudden wind flow blows steadily across the sea surface, physical wave generation process will be like;
1. Turbulent wind flows form random pressure fluctuations at the sea surface. Small waves with a few centimeters order of wavelengths is generated by the pressure fluctuations. (The Phillips mechanism^{[6]})
2. The cross wind keep acting on the initially fluctuated sea surface, then the wave become larger. As the wave become larger, the pressure differences get larger along to the wave growing, then the wave growth rate is getting faster. Then the shear instability expedites the wave growing exponentially. (The Miles mechanism^{[6]})
3. The interactions between the waves on the surface generate longer waves (Hasselmann et al., 1973^{[9]}) and the interaction will transfer wave energy from the shorter waves generated by the Miles mechanism to the waves have slightly lower frequencies than the frequency at the peak wave magnitudes, then finally the waves will be faster than the cross wind speed (Pierson & Moskowitz^{[10]}).
Conditions Necessary for a Fully Developed Sea at Given Wind Speeds, and the Parameters of the Resulting Waves  

Wind Conditions  Wave Size  
Wind Speed in One Direction  Fetch  Wind Duration  Average Height  Average Wavelength  Average Period and Speed 
19 km/h (12 mph)  19 km (12 mi)  2 hr  0.27 m (0.9 ft)  8.5 m (28 ft)  3.0 sec 9.3 ft/sec 
37 km/h (23 mph)  139 km (86 mi)  10 hr  1.5 m (4.9 ft)  33.8 m (111 ft)  5.7 sec 19.5 ft/sec 
56 km/h (35 mph)  518 km (322 mi)  23 hr  4.1 m (13.6 ft)  76.5 m (251 ft)  8.6 sec 29.2 ft/sec 
74 km/h (46 mph)  1,313 km (816 mi)  42 hr  8.5 m (27.9 ft)  136 m (446 ft)  11.4 sec 39.1 ft/sec 
92 km/h (58 mph)  2,627 km (1,633 mi)  69 hr  14.8 m (48.7 ft)  212.2 m (696 ft)  14.3 sec 48.7 ft/sec 
((NOTE: Most of the wave speeds calculated from the wave length divided by the period are proportional to sqrt (length). Thus, except for the shortest wave length, the waves follow the deep water theory described in the next section. The 28 ft long wave must be either in shallow water or between deep and shallow.))
Types of wind waves
Three different types of wind waves develop over time:
 Capillary waves, or ripples
 Seas
 Swells
Ripples appear on smooth water when the wind blows, but will die quickly if the wind stops. The restoring force that allows them to propagate is surface tension. Seas are the largerscale, often irregular motions that form under sustained winds. These waves tend to last much longer, even after the wind has died, and the restoring force that allows them to propagate is gravity. As waves propagate away from their area of origin, they naturally separate into groups of common direction and wavelength. The sets of waves formed in this way are known as swells.
Individual "rogue waves" (also called "freak waves", "monster waves", "killer waves", and "king waves") much higher than the other waves in the sea state can occur. In the case of the Draupner wave, its Script error: No such module "convert". height was 2.2 times the significant wave height. Such waves are distinct from tides, caused by the Moon and Sun's gravitational pull, tsunamis that are caused by underwater earthquakes or landslides, and waves generated by underwater explosions or the fall of meteorites—all having far longer wavelengths than wind waves.
Yet, the largest ever recorded wind waves are common — not rogue — waves in extreme sea states. For example: Script error: No such module "convert". high waves have been recorded on the RRS Discovery in a sea with Script error: No such module "convert". significant wave height, so the highest wave is only 1.6 times the significant wave height.^{[12]} The biggest recorded by a buoy (as of 2011) was Script error: No such module "convert". high during the 2007 typhoon Krosa near Taiwan.^{[13]}
Ocean waves can be classified based on: the disturbing force(s) that create(s) them; the extent to which the disturbing force(s) continue(s) to influence them after formation; the extent to which the restoring force(s) weaken(s) (or flatten) them; and their wavelength or period. Seismic Sea waves have a period of ~20 minutes, and speeds of 760 km/h (470 mph). Wind waves (deepwater waves) have a period of about 20 seconds.
Wave type  Typical wavelength  Disturbing force  Restoring force 

Capillary wave  < 2 cm  Wind  Surface tension 
Wind wave  60–150 m (200–500 ft)  Wind over ocean  Gravity 
Seiche  Large, variable; a function of basin size  Change in atmospheric pressure, storm surge  Gravity 
Seismic sea wave (tsunami)  200 km (125 mi)  Faulting of sea floor, volcanic eruption, landslide  Gravity 
Tide  Half the circumference of Earth  Gravitational attraction, rotation of Earth  Gravity 
The speed of all ocean waves is controlled by gravity, wavelength, and water depth. Most characteristics of ocean waves depend on the relationship between their wavelength and water depth. Wavelength determines the size of the orbits of water molecules within a wave, but water depth determines the shape of the orbits. The paths of water molecules in a wind wave are circular only when the wave is traveling in deep water. A wave cannot "feel" the bottom when it moves through water deeper than half its wavelength because too little wave energy is contained in the small circles below that depth. Waves moving through water deeper than half their wavelength are known as deepwater waves. On the other hand, the orbits of water molecules in waves moving through shallow water are flattened by the proximity of the sea surface bottom. Waves in water shallower than 1/20 their original wavelength are known as shallowwater waves. Transitional waves travel through water deeper than 1/20 their original wavelength but shallower than half their original wavelength.
In general, the longer the wavelength, the faster the wave energy will move through the water. For deepwater waves, this relationship is represented with the following formula:
 <math> C = {L}/{T} </math>
where C is speed (celerity), L is wavelength, and T is time, or period (in seconds).
The speed of a deepwater wave may also be approximated by:
 <math> C = \sqrt{{gL}/{2\pi}} </math>
where g is the acceleration due to gravity, 9.8 meters (32.2 feet) per second squared. Because g and π (3.14) are constants, the equation can be reduced to:
 <math> C = 1.251\sqrt{L} </math>
when C is measured in meters per second and L in meters. Note that in both instances that wave speed is proportional to wavelength.
The speed of shallowwater waves is described by a different equation that may be written as:
 <math> C = \sqrt{gd} = 3.1\sqrt{d} </math>
where C is speed (in meters per second), g is the acceleration due to gravity, and d is the depth of the water (in meters). The period of a wave remains unchanged regardless of the depth of water through which it is moving. As deepwater waves enter the shallows and feel the bottom, however, their speed is reduced and their crests "bunch up," so their wavelength shortens.
Wave shoaling and refraction
As waves travel from deep to shallow water, their shape alters (wave height increases, speed decreases, and length decreases as wave orbits become asymmetrical). This process is called shoaling.
Wave refraction is the process by which wave crests realign themselves as a result of decreasing water depths. Varying depths along a wave crest cause the crest to travel at different phase speeds, with those parts of the wave in deeper water moving faster than those in shallow water. This process continues until the crests become (nearly) parallel to the depth contours. Rays—lines normal to wave crests between which a fixed amount of energy flux is contained—converge on local shallows and shoals. Therefore, the wave energy between rays is concentrated as they converge, with a resulting increase in wave height.
Because these effects are related to a spatial variation in the phase speed, and because the phase speed also changes with the ambient current – due to the Doppler shift – the same effects of refraction and altering wave height also occur due to current variations. In the case of meeting an adverse current the wave steepens, i.e. its wave height increases while the wave length decreases, similar to the shoaling when the water depth decreases.^{[15]}
Wave breaking
Some waves undergo a phenomenon called "breaking". A breaking wave is one whose base can no longer support its top, causing it to collapse. A wave breaks when it runs into shallow water, or when two wave systems oppose and combine forces. When the slope, or steepness ratio, of a wave is too great, breaking is inevitable.
Individual waves in deep water break when the wave steepness—the ratio of the wave height H to the wavelength λ—exceeds about 0.07, so for H > 0.07 λ. In shallow water, with the water depth small compared to the wavelength, the individual waves break when their wave height H is larger than 0.8 times the water depth h, that is H > 0.8 h.^{[16]} Waves can also break if the wind grows strong enough to blow the crest off the base of the wave.
Three main types of breaking waves are identified by surfers or surf lifesavers. Their varying characteristics make them more or less suitable for surfing, and present different dangers.
 Spilling, or rolling: these are the safest waves on which to surf. They can be found in most areas with relatively flat shorelines. They are the most common type of shorebreak
 Plunging, or dumping: these break suddenly and can "dump" swimmers—pushing them to the bottom with great force. These are the preferred waves for experienced surfers. Strong offshore winds and long wave periods can cause dumpers. They are often found where there is a sudden rise in the sea floor, such as a reef or sandbar.
 Surging: these may never actually break as they approach the water's edge, as the water below them is very deep. They tend to form on steep shorelines. These waves can knock swimmers over and drag them back into deeper water.
Science of waves
Wind waves are mechanical waves that propagate along the interface between water and air; the restoring force is provided by gravity, and so they are often referred to as surface gravity waves. As the wind blows, pressure and friction perturb the equilibrium of the water surface and transfer energy from the air to the water, forming waves. The initial formation of waves by the wind is described in the theory of Phillips from 1957, and the subsequent growth of the small waves has been modeled by Miles, also in 1957.^{[17]}^{[18]}
In linear plane waves of one wavelength in deep water, particles near the surface move not plainly up and down but in vertical circles, forwards above and backwards below. As a result, the surface of the water forms not an exact sine wave, but a curtate cycloid with the sharper curves upwards. As (wave height) / wavelength increases, the wave shape becomes more like a cycloid, and when (wave height) = wavelength / π, the wave shape becomes a cycloid, with the cusps upwards. If something such as wind tries to make the wave any higher at that wavelength, the wave shape tries to become a prolate cycloid, which has a loop at each cusp, and for water molecules to follow its usual vertical circles would need water to go through other water, but it cannot, and some of the wave's energy is used up in throwing up spray, which falls back and makes foam along the wave's crest in a line of foam commonly called a "whitecap" or "white horse". Likewise, in a mixture of waves of various lengths moving in various directions and long waves overtaking short waves, as often seen at sea, the same happens if at any time and place the resulting wave motion "goes prolate" and tries to make water go in a raised loop through other water, causing a patch of spray and foam.
When waves propagate in shallow water, (where the depth is less than half the wavelength) the particle trajectories are compressed into ellipses.^{[20]}^{[21]}
As the wave amplitude (height) increases, the particle paths no longer form closed orbits; rather, after the passage of each crest, particles are displaced slightly from their previous positions, a phenomenon known as Stokes drift.^{[22]}^{[23]}
As the depth below the free surface increases, the radius of the circular motion decreases. At a depth equal to half the wavelength λ, the orbital movement has decayed to less than 5% of its value at the surface. The phase speed (also called the celerity) of a surface gravity wave is – for pure periodic wave motion of smallamplitude waves – well approximated by
 <math>c=\sqrt{\frac{g \lambda}{2\pi} \tanh \left(\frac{2\pi d}{\lambda}\right)}</math>
where
 c = phase speed;
 λ = wavelength;
 d = water depth;
 g = acceleration due to gravity at the Earth's surface.
In deep water, where <math>d \ge \frac{1}{2}\lambda</math>, so <math>\frac{2\pi d}{\lambda} \ge \pi</math> and the hyperbolic tangent approaches <math>1</math>, the speed <math>c</math> approximates
 <math>c_\text{deep}=\sqrt{\frac{g\lambda}{2\pi}}.</math>
In SI units, with <math>c_\text{deep}</math> in m/s, <math>c_\text{deep} \approx 1.25\sqrt\lambda</math>, when <math>\lambda</math> is measured in metres. This expression tells us that waves of different wavelengths travel at different speeds. The fastest waves in a storm are the ones with the longest wavelength. As a result, after a storm, the first waves to arrive on the coast are the longwavelength swells.
For intermediate and shallow water, the Boussinesq equations are applicable, combining frequency dispersion and nonlinear effects. And in very shallow water, the shallow water equations can be used.
If the wavelength is very long compared to the water depth, the phase speed (by taking the limit of c when the wavelength approaches infinity) can be approximated by
 <math>c_\text{shallow} = \lim_{\lambda\rightarrow\infty} c = \sqrt{gd}.</math>
On the other hand, for very short wavelengths, surface tension plays an important role and the phase speed of these gravitycapillary waves can (in deep water) be approximated by
 <math>c_\text{gravitycapillary}=\sqrt{\frac{g \lambda}{2\pi} + \frac{2\pi S}{\rho\lambda}}</math>
where
 S = surface tension of the airwater interface;
 <math>\rho</math> = density of the water.^{[24]}
When several wave trains are present, as is always the case in nature, the waves form groups. In deep water the groups travel at a group velocity which is half of the phase speed.^{[25]} Following a single wave in a group one can see the wave appearing at the back of the group, growing and finally disappearing at the front of the group.
As the water depth <math>d</math> decreases towards the coast, this will have an effect: wave height changes due to wave shoaling and refraction. As the wave height increases, the wave may become unstable when the crest of the wave moves faster than the trough. This causes surf, a breaking of the waves.
The movement of wind waves can be captured by wave energy devices. The energy density (per unit area) of regular sinusoidal waves depends on the water density <math>\rho</math>, gravity acceleration <math>g</math> and the wave height <math>H</math> (which, for regular waves, is equal to twice the amplitude, <math>a</math>):
 <math>E=\frac{1}{8}\rho g H^2=\frac{1}{2}\rho g a^2.</math>
The velocity of propagation of this energy is the group velocity.
Wind wave models
Surfers are very interested in the wave forecasts. There are many websites that provide predictions of the surf quality for the upcoming days and weeks. Wind wave models are driven by more general weather models that predict the winds and pressures over the oceans, seas and lakes.
Wind wave models are also an important part of examining the impact of shore protection and beach nourishment proposals. For many beach areas there is only patchy information about the wave climate, therefore estimating the effect of wind waves is important for managing littoral environments.
Seismic signals
Ocean water waves generate land seismic waves that propagate hundreds of kilometers into the land.^{[26]} These seismic signals usually have the period of 6 ± 2 seconds. Such recordings were first reported and understood in about 1900.
There are two types of seismic "ocean waves". The primary waves are generated in shallow waters by direct water waveland interaction and have the same period as the water waves (10 to 16 seconds). The more powerful secondary waves are generated by the superposition of ocean waves of equal period traveling in opposite directions, thus generating standing gravity waves – with an associated pressure oscillation at half the period, which is not diminishing with depth. The theory for microseism generation by standing waves was provided by Michael LonguetHiggins in 1950, after in 1941 Pierre Bernard suggested this relation with standing waves on the basis of observations.^{[27]}^{[28]}
Internal waves
Internal waves can form at the boundary between water layers of different densities. These subsurface waves are called internal waves. As is the case with ocean waves at the airocean interface, internal waves possess troughs, crests, wavelength, and period. Internal waves move very slowly because the density difference between the joined media is very small. Internal waves occur in the ocean at the base of the pycnocline, especially at the bottom edge of a steep thermocline. The wave height of internal waves may be greater than 30 meters (100 feet), causing the pycnocline to undulate slowly through a considerable depth. Their wavelength often exceeds Script error: No such module "convert". and their periods are typically 5 to 8 minutes. Internal waves are generated by wind energy, tidal energy, and ocean currents. Surface manifestations of internal waves have been photographed from space.
Internal waves may mix nutrients into surface water and trigger plankton blooms. They can also affect submarines and oil platforms.
See also
Notes
 ^ Tolman, H.L. (23 June 2010), "Practical wind wave modeling", in Mahmood, M.F., CBMS Conference Proceedings on Water Waves: Theory and Experiment (PDF), Howard University, US, 13–18 May 2008: World Scientific Publications, ISBN 9789814304238
 ^ Holthuijsen (2007), page 5.
 ^ Lorenz, R. D. and A. G. Hayes, The Growth of WindWaves in Titan's Hydrocarbon Seas, Icarus, 219, 468–475, 2012
 ^ Young, I. R. (1999). Wind generated ocean waves. Elsevier. ISBN 0080433170. p. 83.
 ^ Weisse, Ralf; von Storch, Hans (2008). Marine climate change: Ocean waves, storms and surges in the perspective of climate change. Springer. p. 51. ISBN 9783540253167.
 ^ ^{a} ^{b} ^{c} Phillips, O. M. (1957), "On the generation of waves by turbulent wind", Journal of Fluid Mechanics 2 (5): 417–445, Bibcode:1957JFM.....2..417P, doi:10.1017/S0022112057000233
 ^ Miles, J. W. (1957), "On the generation of surface waves by shear flows", Journal of Fluid Mechanics 3 (2): 185–204, Bibcode:1957JFM.....3..185M, doi:10.1017/S0022112057000567
 ^ Chapter 16  Ocean Waves
 ^ Hasselmann K., T.P. Barnett, E. Bouws, H. Carlson, D.E. Cartwright, K. Enke, J.A. Ewing, H. Gienapp, D.E. Hasselmann, P. Kruseman, A. Meerburg, P. Mller, D.J. Olbers, K. Richter, W. Sell, and H. Walden. Measurements of windwave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)' Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe, A(8) (Nr. 12), p.95, 1973.
 ^ Pierson, Willard J., Jr. and Moskowitz, Lionel A. Proposed Spectral Form for Fully Developed Wind Seas Based on the Similarity Theory of S. A. Kitaigorodskii, Journal of Geophysical Research, Vol. 69, p.51815190, 1964.
 ^ Munk, Walter H. (1950), "Origin and generation of waves", Proceedings 1st International Conference on Coastal Engineering, Long Beach, California: ASCE, pp. 1–4
 ^ Holliday, Naomi P.; Yelland, Margaret J.; Pascal, Robin; Swail, Val R.; Taylor, Peter K.; Griffiths, Colin R.; Kent, Elizabeth (2006), "Were extreme waves in the Rockall Trough the largest ever recorded?", Geophysical Research Letters 33 (L05613), Bibcode:2006GeoRL..3305613H, doi:10.1029/2005GL025238
 ^ P. C. Liu; H. S. Chen; D.J. Doong; C. C. Kao; Y.J. G. Hsu (11 June 2008), "Monstrous ocean waves during typhoon Krosa" (PDF), Annales Geophysicae (European Geosciences Union) 26: 1327–1329, Bibcode:2008AnGeo..26.1327L, doi:10.5194/angeo2613272008
 ^ Tom Garrison (2009). Oceanography: An Invitation to Marine Science (7th Edition). Yolanda Cossio. ISBN 9780495391937.
 ^ LonguetHiggins, M.S.; Stewart, R.W. (1964), "Radiation stresses in water waves; a physical discussion, with applications", Deep Sea Research 11 (4): 529–562, doi:10.1016/00117471(64)900014
 ^ R.J. Dean and R.A. Dalrymple (2002). Coastal processes with engineering applications. Cambridge University Press. ISBN 0521602750. p. 96–97.
 ^ Phillips, O. M. (1957), "On the generation of waves by turbulent wind", Journal of Fluid Mechanics 2 (5): 417–445, Bibcode:1957JFM.....2..417P, doi:10.1017/S0022112057000233
 ^ Miles, J. W. (1957), "On the generation of surface waves by shear flows", Journal of Fluid Mechanics 3 (2): 185–204, Bibcode:1957JFM.....3..185M, doi:10.1017/S0022112057000567
 ^ Figure 6 from: Wiegel, R.L.; Johnson, J.W. (1950), "Elements of wave theory", Proceedings 1st International Conference on Coastal Engineering, Long Beach, California: ASCE, pp. 5–21
 ^ For the particle trajectories within the framework of linear wave theory, see for instance:
Phillips (1977), page 44.
Lamb, H. (1994). Hydrodynamics (6th edition ed.). Cambridge University Press. ISBN 9780521458689. Originally published in 1879, the 6th extended edition appeared first in 1932. See §229, page 367.
L. D. Landau and E. M. Lifshitz (1986). Fluid mechanics. Course of Theoretical Physics 6 (Second revised edition ed.). Pergamon Press. ISBN 0080339328. See page 33.  ^ A good illustration of the wave motion according to linear theory is given by Prof. Robert Dalrymple's Java applet.
 ^ For nonlinear waves, the particle paths are not closed, as found by George Gabriel Stokes in 1847, see the original paper by Stokes. Or in Phillips (1977), page 44: "To this order, it is evident that the particle paths are not exactly closed ... pointed out by Stokes (1847) in his classical investigation".
 ^ Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in:
J.M. Williams (1981). "Limiting gravity waves in water of ﬁnite depth". Philosophical Transactions of the Royal Society A 302 (1466): 139–188. Bibcode:1981RSPTA.302..139W. doi:10.1098/rsta.1981.0159.
J.M. Williams (1985). Tables of progressive gravity waves. Pitman. ISBN 9780273087335.  ^ Carl Nordling, Jonny Östermalm (2006). Physics Handbook for Science and Engineering (Eight edition ed.). Studentliteratur. p. 263. ISBN 9789144044538.
 ^ In deep water, the group velocity is half the phase velocity, as is shown here. Another reference is [1].
 ^ Peter Bormann. Seismic Signals and Noise
 ^ Bernard, P. (1941), "Sur certaines proprietes de la boule etudiees a l'aide des enregistrements seismographiques", Bull. Inst. Oceanogr. Monaco 800: 1–19
 ^ LonguetHiggins, M.S. (1950), "A theory of the origin of microseisms", Philosophical Transactions of the Royal Society A 243 (857): 1–35, Bibcode:1950RSPTA.243....1L, doi:10.1098/rsta.1950.0012
References
Scientific
 G.G. Stokes (1880). Mathematical and Physical Papers, Volume I. Cambridge University Press. pp. 197–229.
 Phillips, O.M. (1977). The dynamics of the upper ocean (2nd ed.). Cambridge University Press. ISBN 0521298016.
 Holthuijsen, Leo H. (2007). Waves in oceanic and coastal waters. Cambridge University Press. ISBN 0521860288.
 Janssen, Peter (2004). The interaction of ocean waves and wind. Cambridge University Press. ISBN 9780521465403.
Other
 Rousmaniere, John (1989). The Annapolis Book of Seamanship (2nd revised ed.). Simon & Schuster. ISBN 0671674471.
 Carr, Michael (Oct 1998). "Understanding Waves". Sail: 38–45.
External links
40x40px  Wikimedia Commons has media related to Ocean surface waves. 
40x40px  Wikimedia Commons has media related to Water waves. 
 "Anatomy of a Wave" Holben, Jay boatsafe.com captured 5/23/06
 NOAA National Weather Service
 ESA press release on swell tracking with ASAR onboard ENVISAT
 Introductory oceanography chapter 10 – Ocean Waves
 HyperPhysics – Ocean Waves



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